Momentum Operator in One Dimension

In quantum mechanics, physical observables like momentum are represented by mathematical operators. These operators act on the system's wavefunction to provide information about the observable.

For a particle moving in one dimension (along the x-axis), the momentum operator, often denoted as $\hat{p}_x$, is defined as:

$ \hat{p}_x = -i\hbar \frac{d}{dx} $

Where:

• $i$ is the imaginary unit ($\sqrt{-1}$).
• $\hbar$ is the reduced Planck constant ($h / (2\pi)$).
• $\frac{d}{dx}$ is the differential operator representing the derivative with respect to position $x$.

This operator is fundamental in the formulation of the Schrödinger equation and is used to calculate the momentum expectation value and related properties of a quantum system.

Comparing this definition with the options provided:

• Option 1: $i\hbar \frac{d}{dx}$ is incorrect.
• Option 2: $-i\hbar \frac{d}{dx}$ matches the definition.
• Option 3: $i\hbar \frac{\partial}{\partial t}$ is the energy operator (Hamiltonian) in time-dependent form, not momentum.
• Option 4: $-\frac{\hbar^2}{2m} \frac{d^2}{dx^2}$ is the kinetic energy operator.

Therefore, the correct quantum mechanical operator for the momentum of a particle moving in one dimension is $-i\hbar \frac{d}{dx}$.